Speed of a Travelling Wave: Meaning, Wave Velocity, Applications

Speed of a Travelling Wave: We all know about the fact that light travels faster than sound. Does this mean that sound waves have a speed less than light waves? Can we accurately measure the speed of waves? Waves might have a higher speed in some mediums whereas lower speed in others. So, exactly what factors does the speed of a travelling wave depend on? So, in this article, we will study the speed of a travelling wave and on what conditions does the speed depends.

What is Travelling Wave?

Travelling waves are the type of waves in which the position of particles changes according to the progression of the wave. The maximum amplitude and the minimum amplitude of the wave travel through the medium, and the particles of the wave oscillate according to the wave progression. Travelling waves are also known as progressive waves.

For example, when several particles in a single unidirectional line perform simple harmonic motion in fixed intervals, they form a wave. Due to a small disturbance in a medium, particles in it will start vibrating and will carry energy from one point to another point without any net movement. This makes a wave.

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So, at a particular time, if we view the particles on the wave, it will form a sinusoidal pattern. Hence, all the particles on the wave will have a specific phase difference between them due to the difference in their path.

Here, A is the amplitude of the wave, is the angular frequency of the wave, \(k\) is the propagation constant, \(y\) and \(x\) are displacements of particle and wave respectively and \(t\) is time.

Particles on the wave will have a specific phase difference between them due to the difference in their path.

A travelling wave is represented by the equation:

\(y\, = \,A\,\sin x\sin \,({\bf{\omega }}t\, – \,kx)\)

Wave Velocity

For a wave travelling in a positive \(X\) direction, considering the wave does not change its form, the wave velocity of this wave will be the distance covered by the wave in the direction of propagation per unit time.

Hence, if we consider dx to be the displacement of wave in time \(dt\), we can write for its wave velocity as:

\(v = \,\frac{{dx}}{{dt}}\)

The waveform will retain its displacement hence the phase of the travelling wave must be constant.

\({\bf{\omega }}t = \,kx\, = 0\)

Differentiating the equation on both the sides:

\({\bf{\omega }}\, – \,k\,\frac{{dx}}{{dt}}\, = \,0\)

\(\therefore \,v\, = \,\frac{{\bf{\omega }}}{k}\)

Also, we know for angular frequency \({\bf{\omega }}= 2\pi f\) and wave propagation constant \(k = 2\pi \). Here, \(f\) is the frequency of the wave, and \(\lambda \) is the wavelength of this wave.

\(v = \frac{{2\pi f}}{{\frac{{2\pi }}{\lambda }}}\)

\(\therefore v = f\lambda \)

This equation represents the wave velocity of a travelling wave. Now, let us have a look at the cases of transverse and longitudinal waves.

Types of Travelling Waves

There are two types of travelling waves namely, transverse waves and longitudinal waves.

1. Transverse wave: The kind of travelling wave in which the particles oscillate perpendicular to the direction of propagation of the wave. For example: Electromagnetic waves, waves in a string, etc.
2. Longitudinal wave: The kind of travelling wave in which the particles oscillate along the direction of propagation of wave. For example: Sound waves, heat waves, etc.

Speed of Transverse Wave in a Stretched String

The waves in which the direction of propagation of the wave is perpendicular to that of oscillation of particles are known as transverse waves.

The diagram given shows a string stretched between two rigid supports:

The speed of the transverse wave in the string will depend on two factors:

1. Tension in the string \((T)\)
2. Linear mass density of the string \((\mu )\)

The dimensional formula for tension in the string will be \([T] = [{M^1}{L^1}{T^{ – 2}}]\) the dimensional formula for the linear mass density of the string will be \([\mu ] = [{M^1}{L^{ – 1}}]\)and that of the velocity of the wave will be \([v] = [{L^1}{T^{ – 1}}]\)

Hence, using dimensional analysis,

\(v = c{T^x}{\mu ^v}\)

Here, \(c\) is a dimensionless constant. Applying dimensional formula on both the sides:

\([{L^1}{T^{ – 1}}] = [{M^1}{L^1}\,{T^{ – 2}}]{[{M^1}{L^{ – 1}}]^y}\)

\([{L^1}{T^{ – 1}}]\, =[{M^{x + y}}{L^{x – y}}{T^{ – 2x}}]\)

Comparing both the equations:

\(x + y = 0\)

\(x – y = 1\)

\( – 1 = – 2x\)

Solving these equations;

\(x= \frac{1}{2}, y = – \frac{1}{2}\)

Hence, the speed of the transverse wave in the string will be:

\(v\, = \,c\,\sqrt {\frac{T}{{\rm{\mu }}}} \)

Practically, the value of a dimensionless constant is \(1\). Hence, the speed of the transverse wave in the string is:

\(v\, = \,\,\sqrt {\frac{T}{{\rm{\mu }}}} \)

Speed of Longitudinal Waves in a Medium

The waves in which the direction of propagation of the wave is along the oscillation of particles are known as longitudinal waves.

The speed of a longitudinal wave in a medium depends on:

1. Elastic modulus of the medium \((E)\)
2. Density of the medium \(({\bf{\rho }})\)

The dimensional formula for elastic modulus will be \([E]\, = \,[{M^1}{L^{ – 1}}{T^{ – 2}}]\) the dimensional formula for density will be \([{\bf{\rho }}]\, = \,[{M^1}{L^{ – 3}}]\) and that of the speed of the wave will be \lbrack v\rbrack=\lbrack L^1T^{-1}\rbrack

Hence, using dimensional analysis:

\([v]\, = \,c{E^x}{{\bf{\rho }}^y}\)

Here, c is a dimensionless constant. Applying dimensionless formula on both the sides:

\([{L^1}{T^{ – 1}}] = \,{[{M^1}{L^{ – 1}}{T^{ – 2}}]^x}{[{M^1}{L^{ – 3}}]^y}\)

\([{L^1}{T^{ – 1}}] = \,[{M^{x + y}}{L^{ – x – 3y}}{T^{ – 2x}}]\)

Comparing on both sides:

\(x + y = 0\)

\( – x – 3y = 1\)

\( – 1\, = \, – 2x\)

Solving these equations:

\(x\, = \,\frac{1}{2},\,y\, = \, – \,\frac{1}{2}\)

Hence, the speed of longitudinal wave in the medium will be:

\(v = \,c\,\sqrt {\frac{E}{\rho }} \)

Practically, the value of a dimensionless constant is \(1\). Hence, the speed of a longitudinal wave in the medium is:

\(\therefore v = \,c\,\sqrt {\frac{E}{\rho }} \)

Generally, sound waves are produced by continuously changing pressure in the medium. They travel in the form of compression and rarefaction of the small element of the medium. 

For longitudinal wave in a one-dimensional bar, its elastic modulus is its Young’s modulus which can be equated as:

\(Y = \frac{{{\rm{Longitudinal}}\,{\rm{stress}}}}{{{\rm{Longitudinal}}\,{\rm{strain}}}}\)

Thus, the speed of longitudinal waves in such solid bar will be:

\(v = \sqrt {\frac{\gamma }{\rho }} \)

For a fluid medium, its elastic modulus is known as bulk modulus, which is given by:

\(B\, = \,\frac{{\Delta P}}{{\frac{{\Delta Y}}{V}}}\)

Thus, the speed of longitudinal waves in a fluid medium will be:

\(v = \,\sqrt {\frac{B}{\rho }} \)

These equations were introduced by Newton, and hence they are known as Newton’s equations for the speed of longitudinal waves.

Laplace’s Correction

In the above cases, Newton assumed that the pressure-volume changes in a gas are according to the isothermal process. But according to Laplace, pressure-volume changes occur according to the adiabatic process, and subsequently, an agreement between theoretical and practical observations were made, which favoured Laplace’s observations.

For an adiabatic process, the pressure and volume can change according to the equation:

\(P{V^\gamma } = {\rm{constant}}\)

Differentiating this equation with respect to volume on both the sides:

\(\frac{d}{{dv}}(P{V^\gamma }) = 0\)

\(\therefore \gamma P{V^{\gamma  – 1}} + {V^\gamma }\frac{{dP}}{{dV}} = 0\)

\(\therefore  – \frac{{dP}}{{\frac{{dV}}{V}}} = \gamma P\)

\(\therefore B = \,\gamma P\)

Hence, the speed of longitudinal waves in a gaseous medium will be:

\(v = \sqrt {\frac{{\gamma P}}{\rho }} \)

This equation was known as Laplace’s correction in the speed of longitudinal waves.

Difference between Travelling Waves and Standing Waves

Solved Examples on Speed of Travelling Waves

Q.1: For a travelling wave represented by the equation \(y = 0.1\,\sin \,\sin \,(200\pi t – 4\pi x)\) where \(x\) and \(y\) are in meter, and \(t\) is in second, calculate the wave velocity.
Solution: From the equation, we can obtain the following values:
Angular frequency of the travelling wave \((\omega )\, = \,200\,\pi \,{\rm{rad/s}}\)
Propagation constant of the travelling wave \((k)\, = \,4\pi \,{m^{ – 1}}\)
Velocity of a travelling wave is given by the equation:
\(v = \,\frac{\omega }{k}\)
Substituting the values:
\(v = \,\frac{{200\,\pi }}{{4\pi }}\)
\(\therefore v = \,50\,{\rm{m/s}}\)
The velocity of a travelling wave for the given wave equation is \(50\,{\rm{m/s}}\)

Q.2: A string of uniform linear mass density \({\rm{0}}{\rm{.2}}\,{\rm{g/cm}}\) is hanging vertically through a rigid support using a block of mass \({\rm{100\, g}}\) tied at its other end. Find the speed of transverse waves produced in it on providing a small vibration to it.
Solution: Linear mass density of the string \(\left( \mu  \right) = 0.2\,{\rm{g/cm}} = 0.02\,{\rm{kg/m}}\)
Mass of the block tied to this string \(({\rm{m}})\, = \,100\,{\rm{g}}\,0.1\,{\rm{kg}}\)
Acceleration due to gravity \({\rm{(g)  =  9}}{\rm{.8}}\,{\rm{m/}}{{\rm{s}}^2}\)
Tension in the string due to the mass tied at one end is: 
\(T = \,{\rm{mg}}\)
Substituting the values
\({\rm{T  =  0}}{\rm{.1 }}\, \times \,\,9.8\)
\(\therefore \,T\, = \,0.98\,N\)
Speed of transverse wave in a string will be:
\(v\, = \,\sqrt {\frac{T}{\mu }} \)
Substituting the values,
\(v\, = \,\sqrt {\frac{{0.98}}{{0.02}}} \)
\(\therefore \,\,v = 7\,{\rm{m/s}}\)
Speed of transverse waves in this string is \(7\,{\rm{m/s}}\)

Q.3: The ratio of bulk modulus for two liquid mediums is \(4:1\) and the ratio of its density is \(1:4\) What will be the ratio of the speed of sound waves for both the mediums?
Solution: Ratio of bulk modulus for two liquid media \(({B_1}:{B_2}) = 4:\)
Ratio of their density \(({\rho _1}:{\rho _2})\, = \,1,:4\)
Speed of longitudinal waves in a liquid medium will be given by:
\(v= \sqrt {\frac{B}{\rho }} \)
Hence, we can find the ratio of velocity in sound waves in both the mediums as:
\(\frac{{{v_1}}}{{{v_2}}} = \,\sqrt {\frac{{{B_1}{\rho _2}}}{{{B_2}\,{\rho _1}}}} \)
Substituting the values,
\(\frac{{{v_1}}}{{{v_2}}} = \sqrt {\frac{4}{1} \times \frac{1}{4}} \)
\(\therefore \,{v_1}:{v_2} = 1:1\)
The ratio of sound waves in both the given liquid mediums is \(1:1\)

Applications of Travelling Waves

Travelling waves have become a part of daily lives. We see or feel many travelling waves such as light waves, sound waves, heat waves, etc. Hence, travelling waves have become an important study for day-to-day real-life applications. Some practical life applications of travelling waves are as given below:

1. Electromagnetic waves, alternating current, etc. are a type of travelling waves.
2. Knowing the speed of sound in air, weather can be forecasted.
3. Travelling waves has found a huge application in Ultra-High Voltage transmission systems for the need of high-speed protection. This is also known as Travelling-wave-based protection.

Summary

From this article, we came to know about the travelling wave. How the wave is formed and two different types of travelling waves viz. transverse wave and longitudinal wave. 

Speed of travelling wave: \(v\, = \,f\lambda \)

Speed of transverse wave in a string: \(v\, = \,\sqrt {\frac{T}{\mu }} \)

Speed of longitudinal wave in a medium: \(v\, = \,\sqrt {\frac{F}{\rho }} \)

Frequently Asked Questions (FAQs) on Speed of Travelling Waves

Q.1. What is a travelling wave?
Ans: A travelling wave is a small disturbance in a medium of a stretched string that can carry energy from one point to another without any net movement.

Q.2. How do we find the velocity of a travelling wave?
Ans. The velocity of a travelling wave can be found by knowing the displacement of any point on the wave (for simplicity, we generally use displacement of a point on crest or trough) per unit time.

Q.3. What are the two types of travelling waves?
Ans. The two types of travelling waves are the transverse wave (in which the direction of propagation of wave is perpendicular to the oscillation of particles) and the longitudinal wave (in which the direction of propagation of wave is along the oscillation of particles).

Q.4. Does anything travel faster than light?
Ans. According to Einstein’s special theory of relativity, there is no known object in the universe that can travel faster than light. The estimated speed of light is close to \(299,\,792\,{\rm{km/s}}\).

Q.5. What are the applications of travelling waves?
Ans. Travelling waves have found many applications like medical technology, weather forecasting, fault detection in power plants, etc.

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